Although Theorem \(\ref {binary-quaternary}\)  is stated for linear codes over \(\mathbb {F}_4\) , in general it holds for linear codes over any finite field extension of the binary field; see Theorem 3.8.8 of [1]}.
Next, we give an analogue of Theorem \(\ref {lisonek duadic}\)  to binary cyclic codes that satisfy Theorem \(\ref {binary-quaternary}\) . We denote the Euclidean dual of a binary linear code \(C\)  by \(C^\bot \) .
